# What are the extrema and saddle points of #f(x,y) = xy + 1/x^3 + 1/y^2#?

The point

Since the pure (non-mixed) second-order partial derivatives are also positive, it follows that the critical point is a local minimum.

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To find the extrema and saddle points of ( f(x,y) = xy + \frac{1}{x^3} + \frac{1}{y^2} ), we need to find the critical points by setting the partial derivatives equal to zero and then use the second partial derivative test to determine their nature.

First, find the partial derivatives: [ \frac{\partial f}{\partial x} = y - \frac{3}{x^4} ] [ \frac{\partial f}{\partial y} = x - \frac{2}{y^3} ]

Setting these partial derivatives equal to zero, we get: [ y = \frac{3}{x^4} ] [ x = \frac{2}{y^3} ]

Substituting these into the original function to find the critical points: [ f(x,y) = xy + \frac{1}{x^3} + \frac{1}{y^2} ] [ f\left(\frac{2}{y^3}, y\right) = \frac{2}{y^3}y + \frac{1}{\left(\frac{2}{y^3}\right)^3} + \frac{1}{y^2} ] [ f\left(\frac{2}{y^3}, y\right) = \frac{2}{y^2} + \frac{y^3}{8} + \frac{1}{y^2} ] [ f\left(\frac{2}{y^3}, y\right) = \frac{3}{y^2} + \frac{y^3}{8} ]

Now, to find the extrema and saddle points, we need to analyze the second partial derivatives:

[ \frac{\partial^2 f}{\partial x^2} = \frac{12}{x^5} ] [ \frac{\partial^2 f}{\partial y^2} = \frac{6}{y^4} ] [ \frac{\partial^2 f}{\partial x \partial y} = 1 ]

Using the second partial derivative test, we evaluate ( \frac{\partial^2 f}{\partial x^2} ) and ( \frac{\partial^2 f}{\partial y^2} ) at the critical points. If ( \frac{\partial^2 f}{\partial x^2} \frac{\partial^2 f}{\partial y^2} - \left(\frac{\partial^2 f}{\partial x \partial y}\right)^2 > 0 ), then it's a minimum or maximum. If ( \frac{\partial^2 f}{\partial x^2} ) is positive, it's a minimum, and if it's negative, it's a maximum.

If ( \frac{\partial^2 f}{\partial x^2} \frac{\partial^2 f}{\partial y^2} - \left(\frac{\partial^2 f}{\partial x \partial y}\right)^2 < 0 ), then it's a saddle point.

Substitute the critical points into the second partial derivatives and evaluate the conditions to determine the nature of critical points.

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